Abstract
Estimation and hypothesis test for partial linear single-index multiplicative models are considered in this paper. To estimate unknown single-index parameter, we propose a profile least product relative error estimator coupled with a leave-one-component-out method. To test a hypothesis on the parametric components, a Wald-type test statistic is proposed. We employ the smoothly clipped absolute deviation penalty to select relevant variables. To study model checking problem, we propose a variant of the integrated conditional moment test statistic by using linear projection weighting function, and we also suggest a bootstrap procedure for calculating critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for illustration.
Similar content being viewed by others
References
Bierens, H. J. (1982). Consistent model specification tests. Journal of Econometrics, 20, 105–134.
Bierens, H. J., Ploberger, W. (1997). Asymptotic theory of integrated conditional moment tests. Econometrica, 65, 1129–1151.
Bindele, H. F., Abebe, A., Meyer, K. N. (2018). General rank-based estimation for regression single index models. Annals of the Institute of Statistical Mathematics, 70(5), 1115–1146.
Boente, G., Rodriguez, D. (2012). Robust estimates in generalized partially linear single-index models. Test, 21(2), 386–411.
Chen, K., Guo, S., Lin, Y., Ying, Z. (2010). Least absolute relative error estimation. Journal of the American Statistical Association, 105(491), 1104–1112.
Chen, K., Lin, Y., Wang, Z., Ying, Z. (2016). Least product relative error estimation. Journal of Multivariate Analysis, 144, 91–98.
Cui, X., Härdle, W. K., Zhu, L. (2011). The EFM approach for single-index models. The Annals of Statistics, 39(3), 1658–1688.
Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030–1051.
Fan, J., Peng, H. (2004). Nonconcave penalized likelihood with a diverging number of parameters. The Annals of Statistics, 32(3), 928–961.
Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 58, 71–120.
Jones, L. K. (1987). On a conjecture of Huber concerning the convergence of projection pursuit regression. The Annals of Statistics, 15, 880–882.
Lai, P., Li, G., Lian, H. (2013). Quadratic inference functions for partially linear single-index models with longitudinal data. Journal of Multivariate Analysis, 118, 115–127.
Lee, T. H., White, H., Granger, C. W. J. (2001). Testing for neglected nonlinearity in time series models: A comparison of neural network methods and alternative tests. Journal of Econometrics, 56, 208–229.
Li, G., Peng, H., Dong, K., Tong, T. (2014). Simultaneous confidence bands and hypothesis testing for single-index models. Statistica Sinica, 24(2), 937–955.
Li, G., Lai, P., Lian, H. (2015). Variable selection and estimation for partially linear single-index models with longitudinal data. Statistics and Computing, 25(3), 579–593.
Li, T., Yang, H., Wang, J. L., Xue, L., Zhu, L. (2011). Correction on estimation for a partial-linear single-index model. The Annals of Statistics, 39(6), 3441–3443.
Lian, H., Liang, H. (2016). Separation of linear and index covariates in partially linear single-index models. Journal of Multivariate Analysis, 143, 56–70.
Lian, H., Liang, H., Carroll, R. J. (2015). Variance function partially linear single-index models. Journal of The Royal Statistical Society Series B-statistical Methodology, 77(1), 171–194.
Liang, H., Wang, N. (2005). Partially linear single-index measurement error models. Statistica Sinica, 15(1), 99–116.
Liang, H., Liu, X., Li, R., Tsai, C. L. (2010). Estimation and testing for partially linear single-index models. The Annals of Statistics, 38, 3811–3836.
Lin, D. Y., Wei, L. J., Ying, Z. (2002). Model-checking techniques based on cumulative residuals. Biometrics, 58, 1–12.
Liu, H., Xia, X. (2018). Estimation and empirical likelihood for single-index multiplicative models. Journal of Statistical Planning & Inference, 193, 70–88.
Ma, S., Zhang, J., Sun, Z., Liang, H. (2014). Integrated conditional moment test for partially linear single index models incorporating dimension-reduction. Electronic Journal of Statistics, 8(1), 523–542.
Peng, H., Huang, T. (2011). Penalized least squares for single index models. Journal of Statistical Planning and Inference, 141(4), 1362–1379.
Stute, W. (1997). Nonparametric model checks for regression. The Annals of Statistics, 25, 613–641.
Stute, W., Zhu, L. X. (2002). Model checks for generalized linear models. Scandinavian Journal of Statistics Theory and Applications, 29, 535–545.
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B Methodological, 58(1), 267–288.
Wang, Z., Chen, Z., Wu, Y. (2017). A relative error estimation approach for multiplicative single index model. Journal of Systems Science & Complexity, 30(5), 1160–1172.
Wei, C., Wang, Q. (2012). Statistical inference on restricted partially linear additive errors-in-variables models. Test, 21(4), 757–774.
Xia, Y., Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. Journal of Multivariate Analysis, 97, 1162–1184.
Xia, Y., Tong, H., Li, W. K., Zhu, L. X. (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society, Series B, 64, 363–410.
Xia, Y., Li, W. K., Tong, H., Zhang, D. (2004). A goodness-of-fit test for single-index models. Statistica Sinica, 14, 1–39.
Zhang, J., Zhu, J., Feng, Z. (2018). Estimation and hypothesis test for single-index multiplicative models. Test https://doi.org/10.1007/s11749-018-0586-2.
Acknowledgements
The authors thank the editor, the associate editor and two referees for their constructive suggestions that helped us to improve the early manuscript. Xia Cui is a College Talent Cultivated by Thousand-Hundred-Ten Program of Guangdong Province, and her research was supported by grants from National Natural Science Foundation of China (NSFC) (No. 11471086), Humans and Social Science Research Team of Guangzhou University (No. 201503XSTD) and the Training Program for Excellent Young College Teachers of Guangdong Province (No. Yq201404). Heng Peng’s research was supported in part by CEGR grant of the Research Grants Council of Hong Kong (Nos. HKBU202012 and HKBU 12302615), FRG grants from Hong Kong Baptist University (Nos. FRG2 14-15/064 and FRG2 /16-17/042).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Zhang, J., Cui, X. & Peng, H. Estimation and hypothesis test for partial linear single-index multiplicative models. Ann Inst Stat Math 72, 699–740 (2020). https://doi.org/10.1007/s10463-019-00706-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-019-00706-6